the previous generation
These are the voyages of the Starhip Boobyprize,
her continuing mission:
to deplore strange new worlds,
to shriek at
strange new life and new civilizations,
to cautiously come from where
no sapient creature(4) will come from again.
Captain’s log: stardate Wednesday. The Time Traveller from the alien planet of Ampstedeath has suggested that the Michelson-Morley Experiment is inconclusive, because the Earth might carry the luminiferous ether along with it. Outa-Data has corrected this misapprehension to his own satisfaction, and has also explained Minkowski spacetime, the role of the light cone in determining future and past. The space hurricane has blown itself out, but the starboard warp nacelle has been bent into the shape of a Romulan banana. Mr La Frog believes he can fix it with a ball of string, some sticky tape, and a thousand-ton matter duplicator. Meanwhile an invisible fleet of nanoships is poised off the port bow. Fortunately we can’t see them. Situation normal.
"It seems to me that time is somewhat mutable in this theory," the Time Traveller repeated, to help the reader recall the plot-line.
"That is true," said Outa-Data. "In fact, in 1911 Paul Langevin pointed out a curious feature of special relativity, known as the twin paradox. Suppose that (Fig.3) two twins, Rosencrantz and Guildenstern, are born on Starbase. Rosencrantz stays there all his life, while Guildenstern travels away at nearly lightspeed, and then turns round and comes home again at the same speed. Because of time dilation, only six years (say) have passed in Guildenstern’s frame of reference, whereas 40 years have passed in Rosencrantz’s frame."
"But surely," said the Time traveller, "The situation is symmetric. In Guildenstern’s frame, it is Rosencrantz who seems to travel at nearly the speed of light. So by the same argument, it is Rosencratz who ages less. And that is absurd."
"That’s why people think it’s a paradox," said Pickup. "But it isn’t. Believe me." The tiny Tritovoglian Ambassador sitting on his console dipped its snout politely to indicate assent.
"It only seems paradoxical," expalined Outa-Data, "if you do not actually look at a spacetime diagram, because then you may think that it does not matter which twin is used as the ‘fixed’ frame. But Guildenstern’s motion involves acceleration (positive and negative), while Rosencrantz’s does not and that destroys the apparent symmetry between the two twins. Acceleration is not a relative quantity in Einstein’s theory. Like I said, ‘relativity’ is a silly name."
The Time Traveller shook his head slowly. "But it is only a theory, of course," he said, almost to himself. "Reality is not like that."
"Oh, but it is! The effect was tested back in the late twentieth century, by transporting atomic clocks around the Earth on jumbo jets. Of course, aircraft are so slow compared to light that the time difference observed (and predicted) was only the tiniest fraction of a second."
Figure 3 The Twin Paradox.
"Um," said the Time Traveller. "Jumbo jets? Atomic clocks?"
"Just take my word for it, my friend," growled Woof. "If Mr Outa-Data says it happens, then it happens." He waved his phaser manacingly.
"No, Mr Woof, we must convince him rationally."
"Nothing more rational than a phaser," said Woof.
"Let’s waste him now!" yelled Kert. "Humanely," he added after a pause.
"No, we need a volunteer," said Pickup.
"Hi guys," said Harold Strimmer, walking into the bridge area. "Just checking the litter situation. Oh dear. Captain Pickup, sir, I must inform you that there is a piece of chewing-gum on your console. I will dispose of it." Before anyone could stop him, he dropped the Tritovoglian Ambassador into a large plastic bag full of rubbish.
"I believe we have found our volunteer," said Pickup. "Help the Ambassador out before there is an intergalactic incident, and then get Mr Strimmer suited up. We will leave him on an asteroid, playing the role of Rosencrantz, while the Boobyprize shoots off and back at the speed of light with everybody else on board."
"Asteroid? Ambassador? Hey, mind my litter-bag, you useless little oik "
* * * * * * *
"It really does work," said the Time Traveller.
"Yeah. But we should have left the smeghead a hologrammatic razor. He looks like the abominable snowman in diving gear."
"So indeed ‘the time is out of joint’, to continue Mr Outa-Data’s Shakespearean motif."
"Precisely. So it ought to be possible to exploit the out-of-jointness to make a time machine."
"As I did," said the Time Traveller.
"Yes, but without that ivory device of yours we must use conventional physics, which means relativity. And to do that, we must understand Einstein’s approach to gravity."
"What," asked Counsellor Coi, "does gravity have to do with time travel? Oh, I see..."
"Everything. Though I admit it is not obvious. You see, Einstein invented another theory, called General Relativity, which was a synthesis of Newtonian gravitation and Special Relativity. You know what Newton said about gravity?"
"Naturally," said the Time Traveller. "It is a force that moves particles away from the perfect straight line paths that they would otherwise follow. The force exerted by any particle of matter varies inversely as the square of the distance."
"You mean, there’s a force between Red Frawd and the Boobyprize?" asked the Cat.
"Yes. An attractive force."
"A repulsive force would be more appropriate," said the Cat. "You guys have a strange dress sense. Brush nylon shirts oh boy."
"The paths that particles follow, in the absence of any forces such as gravity, are geodesics. That is, they are shortest paths, they minimize the total distance between their end points. In flat Minkowski spacetime, the analogous relativistic paths minimize the interval instead. The problem is to add in effects of gravity consistently. Einstein incorporated gravity not as an extra force, but as a distortion of the structure of spacetime, which changes the value of the interval. This variable interval between nearby events is called the metric of spacetime. The usual image is to say that spacetime becomes ‘curved’."
"Curved round what?" asked the Cat.
"It is not curved round anything. It is just intrinsically distorted compared to flat spacetime. You might as well ask ‘flat along what?’ about ordinary Euclidean space."
"OK: flat along what?"
"It was a rhetorical question, Cat."
"Then give me a rhetorical answer!"
"I mean, it is just as sensible or silly a question. The curvature is interpreted physically as the force of gravity, and it causes light-cones to deform. One result is ‘gravitational lensing’, the bending of light by massive objects, which Einstein discovered in 1911 and published in 1915. The effect was first observed during an eclipse of the Sun. More recently it has been discovered that some distant quasars very powerful and very distant cosmological objects produce multiple images in telescopes because their light is lensed by an intervening galaxy. Mr Pox: kindly set up a suitable graphic on the ship’s computer."
Fig.4 illustrates this idea by showing a spacelike section of spacetime (that is, one taken at a ‘fixed’ instant of time) near a star. It takes the form of a curved surface that bends downwards to create a circular valley in which the star sits. This spacetime structure is static: it remains exactly the same as time passes. Light follows geodesics across the surface, and is ‘pulled down’ into the hole, because that path provides a short cut. Particles moving in spacetime at sublight speeds behave in the same way. If you look down on this picture from above you see that the particles no longer follow straight lines, but are ‘pulled towards’ the star, whence the Newtonian picture of a gravitational force.
Figure 4 Bending of light by gravity.
"Far from the star," Said Outa-Data, "this spacetime is very close indeed to Minkowski spacetime; that is, the gravitational effect falls off rapidly and soon becomes negligible. Spacetimes that look like Minkowski spacetime at large distances are said to be asymptotically flat. Remember that term: it will be important for making time machines. Most of our own universe is asymptotically flat, because massive bodies such as stars are scattered very thinly."
"Give me some proton torpedoes and I’ll scatter ‘em even thinner," Kert threatened.
"Asymptotically flat along what?" asked the Cat.
Nobody answered either of them.
The Time Traveller digested this information. "So I can give spacetime any form I wish? That sounds implausibly flexible."
"No. When setting up a spacetime, you cannot just bend things any way you like. The metric must obey the Einstein equations, which relate the motion of freely moving particles to the degree of distortion away from ‘flat’ Minkowski spacetime."
"There is a connection between the distribution of masses within the spacetime, and the structure of the spacetime itself," added Pox, even though Outa-Data glared at him. "As if matter creates and moulds its own space and time."
"Precisely," said Outa-Data sourly. "And now I can explain how twentieth century physicists interpreted the phrase ‘time machine’ within the framework of General Relativity."
The Time Traveller’s interest suddenly increased. He was no longer listening just out of politeness.
"A time machine lets a particle or object return to its own past, so its world-line, a timelike curve, must close into a loop. A time machine is just a closed timelike curve, abbreviated to CTC. Instead of asking ‘is time travel possible?’ we ask ‘can CTCs exist?’."
The Time Traveller leaned forward nervously, and his eyes narrowed. "And can they?"
"Well, in flat Minkowski spacetime, they cannot. Forward and backward light cones the future and past of an event never intersect. But they can intersect in other types of spacetime. The simplest example takes Minkowksi spacetime and ‘rolls it up’ into a cylinder (Fig.5). Then the time coordinate becomes cyclic."
Figure 5 A simple example of a spacetime with a CTC.
"You mean history repeats itself, as in Hindu mythology?" said Renault Pickup.
"Sort of. Spacetime repeats; what happens to history depends upon whether you think free will might be in operation. It is a difficult question and one that Einstein’s equations do not address. They just govern the overall coarse structure of spacetime."
"Yuk! Coarse overalls! Next to my sensitive skin? No way, buddy. Next you’ll be telling me stars have flares!"
"Although a cylindrical spacetime looks curved, actually the corresponding spacetime is not curved not in the gravitational sense. When you roll up a sheet of paper into a cylinder, it does not distort. You can roll it out flat again and the paper isn’t folded or wrinkled."
"Flat along what?" said the Cat. "You still haven’t answered."
"A creature that was confined purely to the surface would not notice that the surface had been bent, because distances on the surface would not have changed. In short, the metric a local property of spacetime structure near a given event does not change. What changes is the global geometry of spacetime, its overall topology."
The Time Traveller sighed. "Another new word."
"Topology is a flexible kind of geometry it studies the properties of shapes that persist when the shape is continuously deformed. Like the presence of holes, say, or knots."
"Ah. In my day this was called analysis situs. It was very new."
"Well, now it is very old and very respectable. Rolling up Minkowski spacetime is an example of a powerful topological trick for building new spacetimes out of old ones: cut-and-paste."
"What we doctors call surgery," said Cleverly Blusher.
"And front-line troops like the Kert Rats call butchery," said Kert. "Only we tend to leave out the paste."
"If you can cut pieces out of known spacetimes, and glue them together without distorting their metrics, then the result is also a possible spacetime."
"You are speaking metaphorically, of course."
"Well, until recently I would have agreed with you. But the Federation has set up a new heavy engineering division, and when I describe it as a ‘heavy engineering’ I really do mean heavy. Extremely heavy. But I am getting ahead of myself."
"Like me," said the Time Traveller. They all laughed politely.
"I say ‘distorting the metric’ rather than ‘bending’, for exactly the reason that I say that rolled-up Mikowski spacetime is not curved. I am talking about intrinsic curvature, as experience by a creature that lives in the spacetime; not about apparent curvature as seen in some external representation. Apparent bending of this type is ‘harmless’ it does not actually change the metric. Now, the rolled up version of Minkowski spacetime is a very simple way to prove that spacetimes that obey the Einstein equations can possess CTCs and thus that time travel is not inconsistent with currently known physics. But that does not imply that time-travel is possible."
"I see that. There is a very important distinction between what is mathematically possible and what is physically feasible."
"Yes. A spacetime is mathematically possible if it obeys the Einstein equations. It is physically feasible if it can exist, or could be created, as part of our own universe. Which is where the heavy engineering comes in. Unfortunately for you, there is no reason to suppose that rolled-up Minkowski spacetime is physically feasible: certainly it would be hard to refashion the universe in that form if it wasn’t already endowed with cyclic time. The search for spacetimes that possess CTCs and have plausible physics is a search for more plausible topologies. There are many mathematically possible topologies, but you cannot get to all of them from here."
"As with the Irishman giving directions," said Captain Emeritus Kert.
"I beg your pardon, Captain?"
"It’s an old joke."
"Ah. Yes, I see."
"Flat along what?" said the Cat.
To Be Continued in Episode 4: Red Fraud
Robert Geroch and Gary T. Horowitz, Global structure of spacetimes, in General Relativity: an Einstein Centenary Survey (editors S.W.Hawking and W.Israel), Cambridge University Press, Cambridge 1979, 212-293.
John Gribbin, In Search of the Edge of Time, Bantam Press, New York 1992.
Jean-Pierre Luminet, Black Holes, Cambridge University Press, Cambridge 1992.
Ian Stewart, The real physics of time travel, Analog 114 (January 1994) 106-130.
(4)It has been pointed out to us that this is sapientist. We'll get it right next time.